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| | <asy> | | <asy> |
| − | size(288); | + | size(216); |
| | | | |
| − | pair A, B, C, D;
| + | draw((0,10)--(0,0)--(10,0)); draw((5,10)--(0,0)--(10,5)); |
| − | A = (0,0); B = (5,0); C = (7,4); D = (2,4);
| + | label("$ct$",(0,10),N); label("$x$",(10,0),E); |
| − | draw((-1,0)--(8,0)); draw((-1,4)--(8,4)); draw((-1,-2)--(3,6)); draw((4,-2)--(8,6)); | + | label("$ct'$",(5,10),N); label("$x'$",(10,5),E); |
| − | draw((-1,-4/7)--(8,32/7)); draw((13/2,-2)--(1/2,6));
| |
| | | | |
| − | dot(A); dot(B); dot(C); dot(D); | + | dot((0,3),blue); dot((3,6),blue); draw((0,3)--(3,6),blue); |
| − | label("$A$",A,NW); label("$B$",B+(0,-1/5),S); label("$C$",C,SE); label("$D$",D+(0,1/5),N);
| + | label("$ct_1$",(0,3),W,blue); label("$ct_2'$",(3,6),NW,blue); |
| − | | |
| − | label("$64^\circ$",D+(1/10,-1/2),S); label("$34^\circ$",(-7/10,-11/14));
| |
| − | label("$x$",(5/2,15/4)); label("$y$",(25/4,15/4)); label("$z$",(3,2));
| |
| − | </asy>
| |
| − | | |
| − | <asy>
| |
| − | size(288);
| |
| − | | |
| − | real e = 2.718281828459045235360287471352662497757247093699959574966967627724076630353;
| |
| − | real pi = 3.141592653589793238462643383279502884197169399375105820974944592307816406286;
| |
| − | | |
| − | xaxis(-3,11);
| |
| − | yaxis(-2,12);
| |
| − | | |
| − | real f(real x) { return (x/2 + 1) * (x/2 - 2) * (x/2 - 3); }
| |
| − | | |
| − | draw(graph(f,-3,11)); draw((0,6)--(8,10)); | |
| − | | |
| − | dot((0,6)); dot((8,10)); dot((-2,0)); dot((4,0)); dot((6,0));
| |
| − | label("$A$",(0,6),NW); label("$B$",(8,10),E); label("$(-1,0)$",(-2,0),SE); label("$(2,0)$",(4,0),SW); label("$(3,0)$",(6,0),SE); | |
| − | | |
| − | limits((-3,-3),(11,12),Crop);
| |
| − | </asy>
| |
| − | | |
| − | <asy>
| |
| − | import three; import graph3;
| |
| − | currentprojection=orthographic(4,-4,1);
| |
| − | size(288);
| |
| − | | |
| − | //xaxis3("$A$",0,3);
| |
| − | //yaxis3("$A'$",0,3);
| |
| − | zaxis3("$\hat{\mathbf{n}}$",0,1);
| |
| − | | |
| − | draw((3,0,0)--(0,4,0)--(-3,0,0)--(0,-4,0)--cycle); draw((1,0,1)--(0,0,0)--(1,0,-1)); draw((0,0,0)--(1,0,0)); draw((1,0,1)--(1,0,-1),dashed);
| |
| − | | |
| − | label("$\mathbf{x}$",(1,0,1),E); label("$\mathbf{x}'$",(1,0,-1),E);
| |
| | </asy> | | </asy> |
![[asy] size(216); draw((0,10)--(0,0)--(10,0)); draw((5,10)--(0,0)--(10,5)); label("$ct$",(0,10),N); label("$x$",(10,0),E); label("$ct'$",(5,10),N); label("$x'$",(10,5),E); dot((0,3),blue); dot((3,6),blue); draw((0,3)--(3,6),blue); label("$ct_1$",(0,3),W,blue); label("$ct_2'$",(3,6),NW,blue); [/asy]](http://latex.artofproblemsolving.com/5/a/a/5aa32b4b39ea87966619720471695320b8335aa8.png)
